2005 AMC 10A Problem 17

Question

In the five-sided star shown, the letters

A,

B,

C,

D,

and

E

are replaced by the numbers

3,

5,

6,

7,

and

9,

although not necessarily in this order. The sums of the numbers at the ends of the line segments

AB,

BC,

CD,

DE,

and

EA

form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?

9

10

11

12

13

Each of the five numbers is an endpoint of exactly two of the segments

The five segment sums total

2(3 + 5 + 6 + 7 + 9),

and the middle term of a five-term arithmetic sequence is its mean

Every number is an endpoint of two segments, so the five segment sums total

2(3 + 5 + 6 + 7 + 9) = 60.

The middle term of a five-term arithmetic sequence equals its mean, which is

\dfrac{60}{5} = 12.

Thus, the correct answer is D.

Problem 17 in other years